Starting from a successful model of the \(\pi \)-meson electromagnetic form factor, we calculate a similar form factor, \(F_{K}(Q^{2})\), of the charged K meson for a wide range of the momentum transfer squared, \(Q^{2}\). The only remaining free parameter is to be determined from the measurements of the K-meson charge radius, \(r_{K}\). We fit this single parameter to the published data of the NA-7 experiment which measured \(F_{K}(Q^{2})\) at \(Q^{2}\rightarrow 0\) and determine our preferred range of \(r_{K}\), which happens to be close to recent lattice results. Still, the accuracy in the determination of \(r_{K}\) is poor. However, future measurements of the K-meson electromagnetic form factor at \(Q^{2}\lesssim 5.5\) GeV\(^{2}\), scheduled in Jefferson Laboratory for 2017, will test our approach and will reduce the uncertainty in \(r_{K}\) significantly.

Eur. Phys. J. C
The K -meson form factor and charge radius: linking low-energy data to future Jefferson Laboratory measurements
A. F. Krutov 2
S. V. Troitsky 1
V. E. Troitsky 0
0 D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University , Moscow 119991 , Russia
1 Institute for Nuclear Research of the Russian Academy of Sciences , 60th
2 Samara University , 443086 Samara , Russia
Starting from a successful model of the π -meson electromagnetic form factor, we calculate a similar form factor, FK (Q2), of the charged K meson for a wide range of the momentum transfer squared, Q2. The only remaining free parameter is to be determined from the measurements of the K -meson charge radius, rK . We fit this single parameter to the published data of the NA-7 experiment which measured FK (Q2) at Q2 → 0 and determine our preferred range of rK , which happens to be close to recent lattice results. Still, the accuracy in the determination of rK is poor. However, future measurements of the K -meson electromagnetic form factor at Q2 5.5 GeV2, scheduled in Jefferson Laboratory for 2017, will test our approach and will reduce the uncertainty in rK significantly.
1 Introduction and outline
Quantitative description of particle systems in the
strongcoupling regime remains one of the most challenging
problems of contemporary particle theory. The electromagnetic
structure of light mesons represents an ideal testbed for
various approaches to practical calculations at strong coupling,
including low-energy effective theories of strong
interactions and their connection to Quantum Chromodynamics
(QCD). Not surprisingly, experimental understanding of
various aspects of the meson structure comprises an important
part of the scientific program of the upgraded Jefferson
Laboratory (JLab) [
1
], presently ready for its start. In particular,
the E12-09-011 experiment, scheduled in JLab Hall C for
2017, will attempt to measure the K -meson electromagnetic
form factor, FK (Q2), at the momentum transfer squared up
to Q2 ∼ 5.5 GeV2 [
2
]. In this work, we address some
important implications of this expected result and revisit previous
scarce data on FK (Q2) in the frameworks of a successful
theoretical model.
Some time ago, a model for the electromagnetic form
factor of the charged π meson, Fπ (Q2), has been developed
(see Refs. [
3–5
] for a detailed description), possessing a few
free parameters, fixed in 1998 [
6
] from the experimental data
available at that time. The model predicted subsequent JLab
experimental results on Fπ (Q2) surprisingly well [
7
]
without further tuning of parameters, despite the fact that the
experimentally accessible range of Q2 was extended by an
order of magnitude [
8
]. Moreover, the model with the very
same parameters predicts the correct QCD asymptotics of
Fπ (Q2) at large Q2 [
9,10
]. Subsequently, the model was also
applied to the calculation of electroweak parameters of the
ρ meson [11], for which particular interesting relations have
been obtained. The theoretical grounds for the model include
a relativistic-invariant approximation [
3
] to the instant form
of the relativistic Hamiltonian dynamics (see e.g. Ref. [
12
]),
while the model’s quantitative parameters in the light-quark
sector are fixed from the successful π -meson study [
6
].
The two principal benefits of our model have been
demonstrated in the π -meson study. The first one is its
predictivity: provided the experimental value of the decay constant
is fixed, only one parameter remains to fit the mean square
radius. Any further dependence on the model details, e.g.
in particular the choice of the phenomenological wave
function, is negligible [
6
]. The second advantage is matching with
QCD predictions in the ultraviolet limit: when
constituentquark masses are switched off, as expected at high energies,
the model reproduces correctly not only the functional form
of the QCD asymptotics, but also the numerical coefficient;
see Refs. [
9,10
] for details. To the best of our knowledge, this
is the only available low-energy model reproducing the QCD
limit without introducing additional parameters. Analytical
properties of the pion form factor, as a function on the
complex Q2 plane, obtained in our model, are the same as follows
from the general quantum field theory principles [28]. Since
the model predicted successfully the values of Fπ (Q2) up to
Q2 ∼ 2.5 GeV2, we expect that it can be used for FK (Q2) at
least in the same domain of momentum transfers. Our model
shares its limitations with other constituent-quark models of
mesons: at present, their parameters cannot be consistently
derived from QCD without additional experimental input.
In this paper, the model is applied to the K meson. This
brings two additional parameters into the game, one being
the mass of the strange constituent quark, Ms , and another
describing the interaction in the light-heavy quark system.
In the case of the π meson, the two corresponding
parameters were uniquely determined from two measured
observables, the π -meson decay constant, fπ , and charge radius,
rπ , which determines the form-factor behavior at low Q2.
We will demonstrate below that, for the K meson, the decay
constant, f K , fixes one combination of parameters, while the
charge radius, rK , is known with large uncertainties, which
makes it difficult to use it for fixing the remaining parameter.
We therefore keep it free and obtain a range of the model
parameters consistent with the present data. We address old
measurements of FK (Q2) at Q2 → 0 obtained in the
NA7 experiment at CERN SPS [
13
], which represents the most
precise source of experimental input for determination of rK .
The value of rK obtained in Ref. [
13
], since then extensively
used in numerous experimental and theoretical works, was
estimated from fitting the data points in the pole
approximation. We demonstrate that the range of Q2 studied in Ref. [
13
]
was sufficiently large for deviations from this
approximation to become important. We fit the NA-7 data points with
our exact functions for FK (Q2), which results in a slightly
shifted value of rK . We note that the obtained allowed range
of rK , bounded by the 68% CL agreement with the data and
by the consistency of the model, is in a better agreement
with the recent lattice results [
14
] than the original result of
Ref. [
13
].
Turning to higher energies, we use the constraints on rK
to fix the allowed range of FK (Q2) functions at modestly
large Q2 6 GeV2. We observe that the spread of the
theoretical curves exceeds considerably the expected precision
of the E12-09-011 experiment in JLab. Therefore, within our
approach, the E12-09-011 results might be used not only
to study FK (Q2) at moderate Q2 but also to constrain its
behavior at Q2 → 0 and to further narrow the
experimentally allowed range of rK . This improvement in the value of
a very soft parameter represents an unexpected application
of the coming experiment, aimed presumably at the studies
at much higher momentum transfers, Q2 ∼ (0.5–5.5) GeV2.
The rest of the paper is organized as follows. In Sect. 2,
we describe briefly the model for FK (Q2), with the
emphasis on the differences between the π - and K -meson
models and on the two new parameters we have to introduce.
Sect. 3 addresses the experimental constraints on rK . The
NA-7 data are reanalyzed here in the frameworks of our
model. In Sect. 4, we present the expected FK (Q2)
behavior at larger Q2 and demonstrate how the E12-09-011 JLab
experiment may improve the precision of the rK
measurement. We briefly conclude in Sect. 5 and present a more
detailed description of the model in the Appendix.
2 The K -meson electromagnetic form factor
The approach we used is based on the instant-form Dirac
relativistic Hamiltonian dynamics (see e.g. Ref. [
12
])
supplemented by the relativistic-invariant modified impulse
approximation [
3
]. The form factor of a system of two quarks with
different masses1 has been calculated, within this approach,
in Ref. [
15
]. For completeness, all necessary formulas are
collected in the Appendix.
In general, we need to know five parameters to proceed
with the real numerical calculation. They include the masses
of the two constituent quarks, Mu and Ms ; the parameter of
the two-quark phenomenological wave function, b, with the
physical meaning of the confinement scale; and two
anomalous magnetic moments of quarks, κu and κs . The values
of the latter are fixed in the same way as it was done for
the π meson, that is through the Gerasimov sum rules; see
Ref. [
16
] for details and the appendix for explicit expressions.
The value of κu is, clearly, the same as it was used for the
π meson. We also take advantage of the working model for
Fπ (Q2) where Mu = 0.22 GeV was fixed. The
phenomenological confinement scale, b, may in principle be different for
different systems, and, for the moment, we keep it as a free
parameter, together with Ms .
At this point, we note that we do not vary the shape of the
phenomenological wave function u(k) used in the
calculation, but fix it instead from the π -meson study and leave
only the scale b as a free parameter. In early theoretical
studies of our model [
15,18
], various wave functions have
been used, which, in general, resulted in different
predictions. However, once the model was applied to
phenomenology, the dependence on the wave-function choice was found
negligible, provided the value of the meson decay constant
was fixed [6]. The theoretical systematic uncertainty related
to the choice of the wave-function shape is small compared
to the experimental uncertainties. Note that the weak
dependence of the results of calculation of electromagnetic form
factors of pseudoscalar mesons on the shape of the wave
function of constituent quarks has been pointed out also in
Ref. [
29
].
1 The π meson is well described with Mu = Md .
The choice of the light constituent-quark mass Mu was
determined from a fit to experimental values of Fπ (Q2) at
small Q2 in Ref. [
6
] and confirmed by subsequent
experimental data. It is interesting to note that, with the same value
of Mu , the mass spectrum of light mesons had been
successfully described in Ref. [
30
], within a different framework.
We leave the study of the meson masses in our model for
future work.
Within our approach, we build up a consistent
phenomenologically successful global fit of electroweak properties of
light mesons. The same parameters have been used first for
the best-studied π meson [
6,10
], then for certain properties
of the ρ meson [11]; now we proceed with the K meson.
To fix the two free parameters, b and Mu , for the π meson,
two experimental observables were used. One was the meson
decay constant, fπ , and another was the meson charge radius,
rπ , determined from experimental measurements of Fπ (Q2)
at Q2 → 0. For the K meson, we may equally well use the
decay constant, fK = (0.1562 ± 0.0010) GeV [
17
], to
eliminate one of the parameters. The expression relating f K to the
model parameters was derived in Ref. [
18
] and is presented
in the Appendix.
It would be natural to use the experimental information on
the K -meson charge radius, rK , to fix the single remaining
free parameter and to predict the behavior of FK (Q2) in the
yet unexplored domain of large Q2. However, as we will
see in the next section, this approach is limited by the poor
experimental knowledge of rK .
3 Experimental constraints on the K -meson charge
radius
The form factor FK (Q2) was measured by the NA-7
experiment at CERN SPS, Ref. [
13
]. This measurement of 1986
remains the most recent and the most precise one, and we will
concentrate on its results in what follows.2 Figure 1 presents
the experimental data points. The authors of Ref. [
13
] used
these data to extract the K -meson charge radius by fitting
their data with the pole approximation,
FK (Q2) = 1/ 1 + Q2 r K2 /6 .
We note in passing that a better fit to data points was obtained
in Ref. [
13
] when the condition FK (0) = 1 was not used,
though a departure from this condition is unphysical. This
resulted in the value of r K2 = 0.34 ± 0.05 fm2 (50% CL),
widely used in subsequent studies.
However, one may note that the actual FK (Q2)
function may deviate from the pole approximation already at
2 Inclusion of an earlier measurement of Ref. [
31
] cannot change our
result because of larger error bars and smaller number of data points,
all of which lay farther away from Q2 = 0.
Q2 ∼ 0.1 GeV2, so that corrections to the pole
approximation are already important for the NA-7 data range. This
means that, to determine the derivative of FK (Q2) at Q2 = 0,
and hence rK , one should use either a shorter range of Q2
or a more complicated approximation. The first option fails
because of the insufficient number of data points. Fortunately,
as described above, we can calculate the function FK (Q2)
within our approach. The model has one free parameter which
we use to fit, by means of the usual χ 2 method, the NA-7 data
points with their experimental error bars. To do that, we
consider the two-dimensional parameter space (Ms , b) of the
model; see Fig. 2.
Fixing the value of f K implies a constraint on (Ms , b), so
that a one-parametric space remains (one can see from Fig. 2
that the precision of the experimental value of f K is so good
that its uncertainty may be neglected). The remaining
freedom is therefore parametrized by a line on the (Ms , b) plane;
different points on the line correspond to different values of
rK . Changing Ms and always keeping b(Ms ) to satisfy the
f K constraint, one may fit the experimental data points for
FK (Q2). One may note, however, from Fig. 2, that not all
values of rK may be achieved, provided the f K constraint is
satisfied. Indeed, at r 2 (Ms , b) < 0.39 fm2, the two curves
K
determined by f K (Ms , b) and rK (Ms , b), Fig. 2, have no
intersection points. Since all other parameters beyond Ms
and b are fixed from the π -meson studies and their values are
confirmed experimentally, we have no freedom to change this
picture. Therefore, the limitation in simultaneous description
of fK and rK is a consequence of our requirement of a joint
description of π and K mesons, and not of the construction
Fig. 2 The two parameters for the K -meson form factor. The
experimental value of fK is reproduced on the red dashed line, with the pink
shade representing the experimental uncertainty (this condition leaves
essentially one-dimensional parameter space). Thin black lines
correspond to different values of r K2 , indicated by numbers (in fm2)
of our relativistic model, whose predictivity is manifested in
this way. The value of Ms ≈ 0.27 GeV corresponds to the
lowest achievable rK , which splits the b(Ms ) line into two
branches, so that a larger value of rK may be obtained for
two distinct values of Ms .
In Fig. 3, we illustrate the results of the fit by
presenting the χ 2(rK ) function determined by this method. We note
that the best fit (χ 2 = 13.5 for 14 degrees of freedom)
corresponds to the lowest value of rK , allowed in our approach, and
that
0.39 fm2 ≤ r K2
≤ 0.42 fm2(68% CL, this work).
(1)
Our best-fit curve is also presented in Fig. 1. It is
interesting to note that our best-fit value of r K2 = 0.39 fm2 is in
a better agreement with the recent lattice calculations [
14
],
r K2 ,lattice = 0.380 ± 0.033 fm2, than the best-fit value
derived with the pole approximation. However, we note that,
being derived from the same data, our value for rK shares
similar large statistical uncertainties with the original result, and
both are compatible at 68% CL. It is interesting to compare
our result also with the values obtained from the data
analysis in the frameworks of the chiral perturbation theory [
32
],
2
which vary between rK ,ChPT,min = 0.354 ± 0.071 fm2 and
2
rK ,ChPT,max = 0.431 ± 0.071 fm2: our 68% CL interval for
r K2 is contained in that of Ref. [
32
] for all their assumptions.
4 Large Q2 and the future JLab experiment
The use of the low-Q2 data constrains, by Eq. (1), the
remaining free parameter of the model through the fitting procedure
described in the previous section. In principle, this allows
us to calculate the FK (Q2) function for a large range of Q2
and to make predictions for the future JLab measurements.
This prediction is presented in Fig. 4, where the Q2 range
probed by the E12-09-011 JLab experiment and estimated
uncertainties of the measurement [
27
] are shown. For
comparison, we present also predictions obtained within other
approaches.
Considering Fig. 4, one immediately notes that the
uncertainty in our predictions, determined by the uncertainty in
the rK measurements, exceeds the expected precision of the
experiment and is of order the typical difference between
model predictions. The E12-09-011 experiment would be
able to distinguish between our model and several other ones.
Since these approaches differ in their original assumptions
as well as in approximations being used, future JLab
experiments will be able, in principle, to contribute to the choice
between various models of the description of
nonperturbative dynamics of strong interactions at large and
intermediate distances. In this context, it is instructive to compare
our results with those of Ref. [
24
]. Their approach differs
from ours by the choice of the form of relativistic
dynamics: they use the light-front dynamics. Another difference is
in the approximations: while we use the so-called modified
impulse approximation, see e.g. Ref. [
5
], the conventional
impulse approximation was used in Ref. [
24
]. Our modified
impulse approximation, unlike the conventional one, does not
violate covariance conditions, nor the current conservation.
Another difference with Ref. [
24
], which may be important
at large Q2, is in the Q2-dependence of quark form factors:
we use a renormalization-group inspired logarithmic
function, while in Ref. [
24
], a dipole form is used. Note that the
value of Mu and the expression for the quark radius coincide
in the two works.
In addition, this consideration opens a surprising new
possibility to constrain the low-Q2 behavior of FK (Q2) and
to reduce the experimental uncertainty in rK . Indeed, the
expected error bars of the experiment are smaller than the
spread of FK (Q2) curves predicted in our model. This spread
is determined by the 68% C.L. allowed variations of the
single not firmly fixed parameter of the model, rK , which
determines the behavior of FK (Q2) at Q2 → 0. Hence, limiting
the spread at large Q2 would narrow the allowed range of rK .
Figure 5 illustrates how the measurement of FK (5.5 GeV2)
in JLab would constrain rK in the case that the uncertainty of
the measurement agrees with the estimate of Ref. [
27
]. We
note that the reduced uncertainty in rK transforms, within
our approach, into a more precise knowledge of the model
Fig. 5 The effect of future JLab measurements on the precision of the
rK value. Suppose that JLab finds the value of FK (Q2 = 5.5 GeV2)
shown in the horizontal axis, with the uncertainty given in Ref. [
27
]
(conservative value). Then, within our approach, one would be able to
constrain rK to lay between the lower and upper thick red curves for
this value of FK (the strip with red vertical hatching). For comparison,
rK of Ref. [
13
] is shown by the dashed line (best fit), dark blue shade
(68% CL) and light blue shade (95% CL). The horizontal strip with
orange diagonal hatching represents the lattice result of Ref. [
14
]
parameter; for instance, Ms , which may be interesting from
a theoretical point of view (cf. Fig. 2).
5 Conclusions
In this work, we discussed present and future experimental
data on the K -meson electromagnetic form factor, FK (Q2),
in the context of the model for the electroweak structure
of light mesons based on the relativistic-invariant modified
impulse approximation in the frameworks of the instant-form
relativistic Hamiltonian dynamics. All but two parameters of
the model are fixed from its successful application to the
π -meson form factor in previous studies, where,
experimentally, the model predictions had been confirmed precisely
by a number of subsequent measurements, and theoretically,
the correct QCD asymptotics was reproduced. Of the two
parameters specific for the K -meson case, one combination
is fixed from the K -meson decay constant, f K , while the
remaining one is related to the K -meson charge radius, rK .
The latter is known experimentally with large uncertainties.
We revisited the NA-7 data on FK (Q2) at Q2 0.1 GeV2
and found the rK range allowed by the data within our model.
This range agrees well with the recent lattice results. Still, the
error bars in rK remain large, which makes the predictions
of the model uncertain in the large-Q2 range to be probed
by the coming E12-09-011 experiment in JLab. However,
this suggested an unexpected application of the coming JLab
measurements of FK (Q2), which, despite being performed
at Q2 ∼ (0.5–5) GeV2, would improve the accuracy of the
rK measurements.
Acknowledgements We thank A. Gasparian and G. Huber for
interesting information on the coming JLab K -meson experiments. The work
of AK was supported in part by the Ministry of Education and Science of
the Russian Federation (Grant No. 1394, state task). The work of ST on
the nonperturbative description of strongly coupled QCD bound states
is supported by the Russian Science Foundation, Grant 14-22-00161.
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Appendix A: Formulas for the form-factor calculation
Appendix A.1: The form factor of a system with two
quarks of different masses
This form factor was calculated, in the present approach, in
Ref. [
15
].
The free form factor is given by
√ss
[s2 − 2s(Ms2 + Mu2) + η2][s 2 − 2s (Ms2 + Mu2) + η2]
Bu (s, Q2, s ) + Bs¯(s, Q2, s ) ,
g0(s, Q2, s )
=
Q2(s + s + Q2)
× 2[λ(s, −Q2, s )]3/2
where Mq is the mass of the constituent quark q, η = Mu2 −
M 2,
s
Bs¯(s, Q2, s ) =
f1(s¯)(s + s + Q2 − 2η) cos(ω1 + ω2)
− f2(s¯) Ms ξ(s, Q2, s ) sin(ω1 + ω2) θ (s, Q2, s ),
2
ξ(s, Q2, s )
=
−λ(s, −Q2, s )Ms2 + ss Q2 − ηQ2(s + s + Q2) + Q2η2,
λ(a, b, c) = a2 + b2 + c2 − 2(ab + ac + bc),
f (s¯) =
1
2Ms G(Es¯)(Q2)
4Ms2 + Q2 ;
f2(s¯) = − Ms 4Ms2 + Q2 ,
the Wigner rotation parameters are
ω1 = arctan
ω2 = arctan
where eq are quark charges, fq (Q2) = 1/(1 + log(rq2 Q2/6))
and rq2 = 0.3/Mq2.
The anomalous magnetic moments κq of quarks q are
calculated following Ref. [
16
]. The values of κu and κd¯ should
satisfy [
16
]
eu + κu
ed + κd
= −1.77.
The π -meson form factor depends [
6
] on the sum κu + κd¯,
which has been fixed in Ref. [
6
] from the condition that the
constituent-quark parameters providing a good description
of the data do not depend on the choice of the shape of the
phenomenological wave function: in this way, the value κu +
κd¯ = 0.0268 has been found for Mu = 0.22 GeV, and we use
this value in the present study. Together with Eq. (A.1), this
condition determines κu and κd unambiguously. The value
¯
of κs is determined [
16
] from
(A.1)
eedu++κκud + ed +κd
es +κs
es+κs
1 + ed +κd
2
= 0.42.
This relation, for fixed κu and κ ¯, determines the value of
d
κs up to the choice of the sign at the square root from the
¯
right-hand side. We choose the negative sign because in the
opposite case, the solution gives an unphysically large value
of κs¯. We arrive at the values of κu ≈ −0.01055 and κs¯ ≈
−0.08099, which are used in this study.
The form factor FK (Q2) is given by the double integral,
FK (Q2) =
√ √
d s d s ϕ(k) g0(s, Q2, s ) ϕ(k ),
where
ϕ(k) =
√s(1 − η2/s2) u(k) k,
k =
(s2 − 2s(Ms2 + Mu2) + η2)/4s,
is the phenomenological wave function of the two-quark
system. Following previous studies of the π meson, we choose
the power-law function,
u(k) = N (k2/b2 + 1)−3,
where the normalization N = 16
by the condition
0
∞
dk k2u(k)2 = 1.
Appendix A.2: The K -meson decay constant
The expression for the decay constant f K was calculated, in
the present approach, in Ref. [
18
]. It reads
2/(7π b3) is determined
f K =
where
G0(s) =
∞
Ms +Mu
√
3
2√2π √s
×
pq0 =
Mq2 + k2.
√
d s G0(s)ϕ(s),
( ps0 + Ms )( pu0 + Mu )
k2
1 − ( ps0 + Ms )( pu0 + Mu )
,
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